Verify that |r^{t}(s)|=1.Consider the helix represented investigation by the vector-valued function r(t)= < 2 cos t, 2 sin t, t >.

Question
Verify that $$\|r^{t}(s)\|=1$$.Consider the helix represented investigation by the vector-valued function $$r(t)=\ <\ 2\ \cos\ t,\ 2\ \sin\ t,\ t\ >$$.

2020-10-29
Given: The function $$\underline{r(t)=\ <\ 2\ \cos\ t,\ 2\ \sin\ t,\ t\ >}$$ Proofs: The curve in terms of arc length is, $$r(s)=2\ \cos\ \left(\frac{s}{\sqrt{5}}\right)i\ +\ 2\ \sin\ \left(\frac{s}{\sqrt{5}}\right)j\ +\ \frac{s}{\sqrt{5}}k$$. On differenting the vector-value function r(s), we get $$r'(s)=\ -\frac{2}{\sqrt{5}}\ \sin \left(\frac{s}{\sqrt{5}}\right)i\ +\ \frac{2}{\sqrt{5}}\ \cos \left(\frac{s}{\sqrt{5}}\right)j\ +\ \frac{1}{\sqrt{5}}k$$ From this calcute $$\|r'(s)\|$$ as $$\|r'(s)\|=\ \sqrt{\left(-\frac{2}{\sqrt{5}}\ \sin\left(\frac{s}{\sqrt{5}}\right)\right)^{2}\ +\ \left(\frac{2}{\sqrt{5}}\ \cos\left(\frac{s}{\sqrt{5}}\right)\right)^{2}\ +\ \left(\frac{1}{\sqrt{5}}\right)^{2}}$$
$$=\ \sqrt{\frac{4}{5}\ +\ \frac{1}{5}}$$
$$=\ \sqrt{\frac{5}{5}}$$
$$= 1$$ Hence, it is proved that $$\|r'(s)\|=1$$

Relevant Questions

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